The metallicity dependence of the long-duration gamma-ray burst rate from host galaxy luminosities

We investigate the difference between the host galaxy properties of core-collapse supernovae (CC SNe) and long-duration gamma-ray bursts (LGRBs), and quantify a possible metallicity dependence of the efficiency of producing LGRBs. We use a sample of 16 CC SNe and 16 LGRBs from Fruchter et al. which have similar redshift distributions to eliminate galaxy evolution biases. We make a forward prediction of their host galaxy luminosity distributions from the overall cosmic metallicity distribution of star formation. The latter is based on luminosity functions, star formation rates (SFRs) and luminosity-metallicity (L-Z) relations of galaxies. This approach is supported by the finding that LGRB hosts follow the L-Z relations of star-forming galaxies. We then compare predictions for metallicity-dependent event efficiencies with the observed host data. We find that ultraviolet-based SFR estimates predict the host distribution of CC SNe perfectly well in a metallicity-independent form. In contrast, LGRB hosts are on average fainter by one magnitude, almost as faint as the Large Magellanic Cloud. Assuming this to be a metallicity effect, the present data are insufficient to discriminate between a sharp cut-off and a soft decrease in efficiency towards higher metallicity. For a sharp cut-off, however, we find a best value for the cut-off metallicity, as reflected in the oxygen abundance, 12 + log (O/H)(lim) similar or equal to 8.7 +/- 0.3 at 95 per cent confidence including systematic uncertainties, in the calibration of Asplund, Grevesse & Sauval. This value is somewhat lower than the traditionally quoted value for the Sun, but is comparable to the revised solar oxygen abundance. LGRB models that require sharp metallicity cut-offs well below approximately one-half the revised solar metallicity appear to be effectively ruled out, as they would require fainter LGRB hosts than those that are observed. We also discuss the likely implications of the still ongoing metallicity 'calibration debate'.